Complex numbers rules

[coarse sage]

Hey,
We recently started taking complex numbers and I just want to know if all the rules we have previously taken apply to complex numbers too or no?
Example:

• dot product of vectors (assume I have a vector with complex coordinates)
• pyth theorem
• cosine and sine law

[shut lantern]

what are cosine and sine law?

the dot product for complex vectors is slightly different

for real-valued vectors it's $$v \cdot w = \sum_i v_i w_i$$

[glacial crypt]

siema! (he/him)

[shut lantern]

but for complex-valued vectors it's $$v \cdot w = \sum_i \overline{v_i} w_i$$

[glacial crypt]

siema! (he/him)

[shut lantern]

so you have to take some complex conjugates

this ensures that (v \cdot v) is equal to (|v|^2)

[glacial crypt]

siema! (he/him)

[coarse sage]

@shut lantern "properties that work in triangles"

[shut lantern]

are you plotting the real numbers on the x axis and the imaginary numbers on the y axis?

[coarse sage]

Yeah

[shut lantern]

Then yeah everything that works in R^2 works in C as well

[coarse sage]

Is geometry an exception?

The rules of geometry*

[shut lantern]

No

The triangle rules will still work

[coarse sage]

But not Pythagorean

[shut lantern]

the pythagorean theorem still works